Structure tensor

These conditions show us immediately that in the case of the four-neighbor HPP lattice (V = 4) f is noni.sotropic, and the macroscopic equations therefore cannot yield a Navier-Stokes equation. For the hexagonal FHP lattice, on the other hand, we have V = 6 and P[. is isotropic through order Wolfram [wolf86c] predicts what models are conducive to f lavier-Stokes-like dynamics by using group theory to analyze the symmetry of tensor structures for polygons and polyhedra in d-dimensions. 

With this tensor structure of the kernel, 2D Laplace inversion can be performed in two steps along each dimension separately [50]. Even though such procedure is applicable when the signal-to-noise ratio is good, the resulting spectrum, however, tends to be noisy [50]. Furthermore, it is not dear how the regularization parameters should be chosen. 

If V(r, x) were a known function, this linear expansion could be used to determine how the velocity varies for short intervals of time and in any arbitrary short spatial direction dx. In a Taylor-series expansion of a scalar field, it is often conventional to post-multiply by the dx. Since the gradient of a scalar field is a vector and because the inner product of two vectors is commutative, the order of the product is unimportant. However, because of the tensor structure of the gradient of a vector field, the pre-multiply is essential. 

Though written in scalar form, the tensor structure of the equation was retained. 

Focusing first on the dominant two-level term [65], careful analysis of the tensor structure, with respect to the proper signs for the damping corrections and utilizing the freedom to add a j,k index-antisymmetric term (see later), yields the following result [59,66] ... 

The next step is the actual renormalization procedure. The crucial observation for both the physical interpretation as well as the technical success of this step is the fact that the divergent contributions to the three relevant functions have the same structure as the corresponding free propagators and the free vertex The divergent part of is just proportional to and m, but not e.g. to p, the divergent part of repeats the tensor structure of D y, Eq.(203), and the divergent part of F is proportional to the free vertex 7 (but does not... 

Tensor Structure of the Many-Electron Hamiltonian and Wave Function 283 8.4... 

The TDQS tensor structure over MQOS has also been studied in deep, providing simple three element MQOS examples to observe in this schematic, but sufficiently illustrative case, all the possible characteristics this kind of discrete molecular representations can have. Also a handy numerical example of this cardinality kind has been chosen to fill up this three-element MQOS scheme the set of three degenerate p-type GTO functions. Such an example not only fulfilled a proper TDQS scheme, associated to a MQOS with cardinality three, but permitted to visualize how QS could handle degenerate QO states. 

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